Previous |  Up |  Next

Article

Full entry | PDF   (0.2 MB)
Keywords:
generalized quantifier; Ramsey theory
Summary:
Let $\binom{I}{m}$ be the set of subsets of $I$ of cardinality $m$. Let $f$ be a coloring of $\binom{I}{m}$ and $g$ a coloring of $\binom{I}{m}$. We write $f\rightarrow g$ if every $f$-homogeneous $H\subseteq I$ is also $g$-homogeneous. The least $m$ such that $f\rightarrow g$ for some $f:\binom{I}{m}\rightarrow k$ is called the {\sl $k$-width} of $g$ and denoted by $w_k(g)$. In the first part of the paper we prove the existence of colorings with high $k$-width. In particular, we show that for each $k>0$ and $m>0$ there is a coloring $g$ with $w_k(g)=m$. In the second part of the paper we give applications of wide colorings in the theory of generalized quantifiers. In particular, we show that for every monadic similarity type $t=(1,\ldots,1)$ there is a generalized quantifier of type $t$ which is not definable in terms of a finite number of generalized quantifiers of a smaller type.
References:
[1] Alon N.: Personal communication, via J. Spencer.
[2] Hella L., Luosto K., Väänänen J.: The Hierarchy Theorem for generalized quantifiers. to appear in the Journal of Symbolic Logic. MR 1412511
[3] Kolaitis Ph., Väänänen J.: Pebble games and generalized quantifiers on finite structures. Annals of Pure and Applied Logic 74 (1995), 23-75; Abstract in {\sl Proc. 7th IEEE Symp. on Logic in Computer Science}, 1992. MR 1336413
[4] Lesniak-Foster L., Straight H.J.: The chromatic number of a graph. Ars Combinatorica 3 (1977), 39-46. MR 0469805
[5] Lindström P.: First order predicate logic with generalized quantifiers. Theoria 32 (1966), 186-195. MR 0244012
[6] Luosto K.: Personal communication.
[7] Mostowski A.: On a generalization of quantifiers. Fundamenta Mathematicae 44 (1957), 12-36. MR 0089816 | Zbl 0078.24401
[8] Nešetřil J.: Ramsey Theory. In: {\sl Handbook of Combinatorics}, (ed. R.L. Graham, M. Grötschel, L. Lovász), North-Holland, 1995. MR 1373681
[9] Nešetřil J., Rödl V.: A simple proof of Galvin-Ramsey property of all finite graphs and a dimension of a graph. Discrete Mathematics 23 (1978), 49-55. MR 0523311
[10] Nešetřil J., Rödl V.: A structural generalization of the Ramsey theorem. Bull. Amer. Math. Soc. 83 (1977), 127-128. MR 0422035
[11] Shelah S.: A finite partition theorem with double exponential bounds. to appear in {\sl Mathematics of Paul Erdös} (ed. R.L. Graham and J. Nešetřil), Springer Verlag, 1996. MR 1425218
[12] Väänänen J.: Unary quantifiers on finite structures. to appear.
[13] Westerståhl D.: Personal communication.

Partner of